Do students pay zakat as well, as long as they have met the minimum Nisab threshold?

Yes. Students do have to pay Zakat if they are above the Nisab level. As long as someone's net Zakatable assets are above Nisab, they pay Zakat. Example: A student could have 10,000 cash and a 10,000 student loan. The loan is however not due until much later and he/she is making no payments toward it yet. So as long as this student has held 10,000 in their bank account for 1 full year, then they must pay Zakat on it (or on any portion of this that is unspent for 1 year).